Geodesic
The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

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Based on the Hermite–Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach. 
@article{DMTCS_2015_special_285_a54,
     author = {Yang, Arthur L.B. and Zhang, Philip B.},
     title = {The {Real-rootedness} of {Eulerian} {Polynomials} via the {Hermite{\textendash}Biehler} {Theorem}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2510},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2510/}
}
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Yang, Arthur L.B.; Zhang, Philip B. The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2510. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2510/

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